Holography principle and arithmetic of algebraic curves

Holography Principle and Arithmetic of Algebraic Curves

According to the holography principle (due to G. ‘t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which ...

Modular Curves and Arithmetic

for relatively prime integers a and N, generate subgroups of finite index in the group of units of number fields F which are abelian over Q. The interest in circular units stems from the fact that theyare "sufficiently many" explicitly constructed units, and also from the fact that their "position" in the full group of global units (and in certain groups of local units) bears upon deep arithmet...

Uncertainty Relation from Holography Principle

We propose that the information and entropy of an isolated system are two sides of one coin in the sense that they can convert into each other by measurement and evolution of the system while the sum of them is identically conserved. The holographic principle is reformulated in the way that this conserved sum is bounded by a quarter of the area A of system boundary. Uncertainty relation is deri...

Arithmetic of Hyperelliptic Curves

Generalized Uncertainty Principle, Extra-dimensions and Holography

We consider Uncertainty Principles which take into account the role of gravity and the possible existence of extra spatial dimensions. Explicit expressions for such Generalized Uncertainty Principles in 4+n dimensions are given and their holographic properties investigated. In particular, we show that the predicted number of degrees of freedom enclosed in a given spatial volume matches the holo...